Why are Venn diagrams useful?

Geeks and Nerds: a cartoon from XKCD (click on the image for a link to this wonderful site)

Venn diagrams works only for two or three sets, but not for four because 4 circles in the plane do not divide the plane in 16 regions. Venn diagrams are useful only because axioms of Boolean algebra can be written using only three variables, thus allowing for a diagrammatic representation of each axiom. I have made a similar observation elsewhere: Coxeter groups allow a powerful level of visualisation because all relations between their canonical generators are in some explicit mathematical sense two dimensional.

A later addition in response to a comment: If we do not insist on sets being represented by circles, then Venn diagrams (although increasingly non-intuitive) can be drawn for larger numbers of sets:

Venn diagrams do have some educational value though. The regions demarcated by the ellipses are the atoms in the free Boolean algebra on generators, and finite unions of those regions give all elements of the free Boolean algebra. It’s nice to have concrete pictures of formal facts like these.

Venn’s novel contribution, and why we remember him still, was to include a boundary quadrilateral around all the circles, enabling visual representation of the complement of a set. No one had done this before him, and he did it at a time when there was great discussion and contention among statisticians about the appropriate definition of “a population” in undertaking statistical inference.

Personally, I would not call your second diagram a Venn diagram, since the diagram border is not explicitly drawn.

Hemant asks “whats in mathematics without problems”. In fact, mathematics is the science of structure and pattern. Problems are a distraction from this, and from the subject’s main purpose. Let’s stop teaching math through problems, and let’s ditch the macho competitiveness that an unwarranted focus on problem-solving has led us to.